1998 IMO Problems/Problem 2
Problem
In a competition, there are contestants and judges, where is an odd integer. Each judge rates each contestant as either “pass” or “fail”. Suppose is a number such that, for any two judges, their ratings coincide for at most contestants. Prove that .
Solution
Let stand for the number of judges who pass the th candidate. The number of pairs of judges who agree on the th contestant is then given by
\begin{align*} {c_i \choose 2} + {{b - c_i} \choose 2} &= \frac{1}{2}\left(c_i(c_i - 1) + (b - c_i)(b - c_i - 1) \right) \\ &= \frac{1}{2}\left( c_i^2 + (b - c_i)^2 - b \right) \\ &\geq \frac{1}{2}\left( \frac{b^2}{2} - b \right) \\ &= \frac{b^2 - 2b}{4} \end{align*}
where the inequality follows from AM-QM. Since is odd, is not divisible by and we can strengthen the inequality to
Letting stand for the number of instances where two judges agreed on a candidate, it follows that
The given condition on implies that
Therefore
which simplifies to
See Also
1998 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |